Originally Posted by MuddauberE
I would defiantly check the short and get that fixed. Shorts are fire hazards. But as far as the different voltage, a 50 AMP service had two main power sources; thus the need for 4 wires instead of three. Many times one of those lines coming in is supplying a lower voltage than the other. Figure out which power ports these are in the receptical and test them. It may not be the trailer, it may be the power source.
Something else is to Try unplugging and running off battery and see if it's still Dim.
You're right that the 50A service is two legs, but usually only one of those legs powers the converter, which supplies the batteries so all the 12v should be only on one leg.
I'm seeing two possible issues here.
1) There's loss in the lines between the batteries and the lights, resulting in one being less than the other.
2) Both are less than the converter should be outputting. I'd start at the battery and check to make sure your converter's supplying a 13+ charge to the batteries. If not, as the batteries drain you'll have less voltage and therefore more subject to potential line loss.
Once your sure that's working, the next thing to check is why is there different line loss between the lights. This could be due to different gauge wires being used, a short, an extremely long run of wiring or too much load downstream on the circuit.